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On locally AH algebras
About this Title
Huaxin Lin, The Research Center for Operator Algebras, Department of Mathematics, East China Normal University, Shanghai, 20062, China — and — Department of Mathematics University of Oregon, Eugene, Oregon 97405
Publication: Memoirs of the American Mathematical Society
Publication Year:
2015; Volume 235, Number 1107
ISBNs: 978-1-4704-1466-5 (print); 978-1-4704-2225-7 (online)
DOI: https://doi.org/10.1090/memo/1107
Published electronically: October 1, 2014
MSC: Primary 46L35, 46L05
Table of Contents
Chapters
- 1. Introduction
- 2. Preliminaries
- 3. Definition of ${\mathcal C}_g$
- 4. $C^*$-algebras in ${\mathcal C}_g$
- 5. Regularity of $C^*$-algebras in ${\mathcal C}_1$
- 6. Traces
- 7. The unitary group
- 8. ${\mathcal Z}$-stability
- 9. General Existence Theorems
- 10. The uniqueness statement and the existence theorem for Bott map
- 11. The Basic Homotopy Lemma
- 12. The proof of the uniqueness theorem 10.4
- 13. The reduction
- 14. Appendix
Abstract
A unital separable $C^*$-algebra $A$ is said to be locally AH with no dimension growth if there is an integer $d>0$ satisfying the following: for any $\epsilon >0$ and any compact subset ${\mathcal F}\subset A,$ there is a unital $C^*$-subalgebra $B$ of $A$ with the form $PC(X, M_n)P,$ where $X$ is a compact metric space with covering dimension no more than $d$ and $P\in C(X, M_n)$ is a projection, such that \[ \operatorname {dist}(a, B)<\epsilon \,\,\,\text {for all}\,\,\,a\in {\mathcal F}. \] We prove that the class of unital separable simple $C^*$-algebras which are locally AH with no dimension growth can be classified up to isomorphism by their Elliott invariant. As a consequence unital separable simple $C^*$-algebras which are locally AH with no dimension growth are isomorphic to a unital simple AH-algebra with no dimension growth. In fact, we show that every unital amenable separable simple $C^*$-algebra with finite tracial rank which satisfies the UCT has tracial rank at most one. Therefore, by the author’s previous result, the class of those unital separable simple amenable $C^*$-algebras $A$ satisfying the UCT which have rationally finite tracial rank can be classified by their Elliott invariant. We also show that unital separable simple $C^*$-algebras which are “tracially" locally AH with slow dimension growth are ${\mathcal Z}$-stable.- Bruce Blackadar, $K$-theory for operator algebras, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986. MR 859867
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